US9128875B2 - Signal transformation apparatus applied hybrid architecture, signal transformation method, and recording medium - Google Patents
Signal transformation apparatus applied hybrid architecture, signal transformation method, and recording medium Download PDFInfo
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- US9128875B2 US9128875B2 US13/537,346 US201213537346A US9128875B2 US 9128875 B2 US9128875 B2 US 9128875B2 US 201213537346 A US201213537346 A US 201213537346A US 9128875 B2 US9128875 B2 US 9128875B2
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- H—ELECTRICITY
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- H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
- H04N19/42—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals characterised by implementation details or hardware specially adapted for video compression or decompression, e.g. dedicated software implementation
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- G—PHYSICS
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- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
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- H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
- H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
- H04N19/102—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or selection affected or controlled by the adaptive coding
- H04N19/12—Selection from among a plurality of transforms or standards, e.g. selection between discrete cosine transform [DCT] and sub-band transform or selection between H.263 and H.264
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- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
- H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
- H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
- H04N19/134—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or criterion affecting or controlling the adaptive coding
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- H—ELECTRICITY
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- H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
- H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
- H04N19/60—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding
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- H04N19/60—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding
- H04N19/63—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding using sub-band based transform, e.g. wavelets
- H04N19/635—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding using sub-band based transform, e.g. wavelets characterised by filter definition or implementation details
Definitions
- the present invention relates generally to a signal transformation apparatus, a signal transformation method, and a recording medium. More particularly, the present invention relates to a signal transformation apparatus with a hybrid architecture applied, a signal transformation method, and a recording medium.
- HEVC High Efficiency Video Coding
- MPEG Moving Picture Experts Group
- VCEG ITU-T Video Coding Experts Group
- JCT-VC Joint Collaborative Team on Video Coding
- DCT Discrete Cosine Transform
- III Discrete Cosine Transform
- Discrete orthogonal transform is used for applications of signal classification and representation.
- Discrete signal processing of Discrete Fourier Transform (DFT) is a popular transformation for Orthogonal Frequency Division Multiplexing 4 th Generation (OFDM-4G) and communication.
- the OFDM is a key technology for next-generation mobile communication (3GPP-LTE, mobile WiMAX, IMT-Advanced) as well as wireless LAN (IEEE 802.11a, IEEE 802.11n), wireless PAN (multiband OFDM), and DFT-based broadcasting (DAB, DVB, DMB). Further, discrete wavelet transform based on Haar Wavelet Transform (HWT) is very useful in JPEG 2000 standard and signal analysis. What is needed is a method for applying the four transforms more easily.
- a primary aspect of the present invention to provide a signal transformation apparatus for selecting one of DCT-II, DST-II, DFT, and HWT, transforming and outputting an input signal according to the selected transform, a signal transformation method, and a recording medium.
- a signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT) includes a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and a transformation part for transforming and outputting an input signal according to the transform selected by the switching part.
- the switching part may include a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT.
- the submatrix selector may select [L] N as a submatrix and the permutation matrix selector selects [D] N as a permutation matrix.
- the transformation part may apply the DCT-II to the input signal by computing Equation (27) using the submatrix [L] N and the permutation matrix [D] N :
- the submatrix selector may select [U] N as a submatrix and the permutation matrix selector selects [D] N as a permutation matrix.
- the transformation part may apply the DST-II to the input signal by computing Equation (45) by using the submatrix [U] N and the permutation matrix [D] N and multiplying every output by a matrix [M 1 ] [M 2 ]:
- the submatrix selector may select [Pr] N as a submatrix and the permutation matrix selector selects [W] n as a permutation matrix.
- the transformation part may apply the DFT to the input signal by computing Equation (56) by using the submatrix [Pr] N and the permutation matrix [W] n , multiplying the input signal by a matrix [M 3 ] during first h steps of 2h steps, and multiplying a matrix [M 4 ] during last h steps:
- the submatrix selector may select [Pa] N as a submatrix and the permutation matrix selector selects [Pb] N as a permutation matrix.
- the transformation part may apply the HWT to the input signal by computing Equation (64) by using the submatrix [Pa] N and the permutation matrix [Pb] N and multiplying the input signal by a matrix [M 5 ] during last h steps of 2h steps:
- a signal transformation method may be applied to the signal transformation apparatus.
- a computer-readable recording medium may contain a computer program for executing functions of the signal transformation apparatus.
- FIG. 1A-1C is a butterfly data flow graph of Equation (27) according to an exemplary embodiment of the present invention.
- FIG. 2A-2C is a butterfly data flow graph of Equation (45) according to an exemplary embodiment of the present invention.
- FIG. 3A-3C is a butterfly data flow graph of Equation (56) according to an exemplary embodiment of the present invention.
- FIG. 4 is a butterfly data flow graph of Equation (64) according to an exemplary embodiment of the present invention.
- FIG. 5 is a block diagram of a signal transformation apparatus adopting a hybrid architecture according to an exemplary embodiment of the present invention
- FIG. 6A to 6C is a detailed block diagram of the signal transformation apparatus according to an exemplary embodiment of the present invention.
- FIG. 6D is a graph of variance comparison of the KLT/DCT/DST/DFT/HWT
- FIG. 7A is a diagram of a one-way video coding scenario according to an exemplary embodiment of the present invention.
- FIG. 7B is a diagram of a two-way video coding scenario according to an exemplary embodiment of the present invention.
- Exemplary embodiments of the present invention provide a fast DCT-II/DST-II/DFT/HWT hybrid transform architecture for new digital videos and fusion mobile handsets based on Jacket-like sparse matrix decomposition.
- the fast hybrid architecture includes source coding standard such as MPEG4 and JPEG 2000, and digital filtering discrete Fourier transform.
- the fast hybrid architecture includes two operations. One operation is Block-wise Inverse Jacket Matrix (BIJM) for DCT-II/DST-II and the other operation is Element-wise Inverse Jacket Matrix (EIJM) for DFT/HWT.
- BIOS Block-wise Inverse Jacket Matrix
- EIJM Element-wise Inverse Jacket Matrix
- the present invention provides sparse unified matrix factorization for the unified chip based on the Jacket matrix.
- DCT-II, DST-II, DFT and HWT matrices can be decomposed to one orthogonal character matrix and a special sparse matrix.
- the inverse of the sparse matrix is a block-wise inverse or an element-wise inverse.
- a DCT-II transform matrix is given as follows.
- a 2 ⁇ 2 matrix which is a basic matrix is given by
- ⁇ ⁇ 1 2 ( 2 ) can be seen as a special element-wise inverse matrix of order 1, and its inverse is ⁇ square root over (2) ⁇
- C l i cos(i ⁇ /l) is the cosine unit for DCT computations.
- the 4-by-4 DCT-II matrix has the form as
- [ Pc ] 4 [ 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ]
- [Pc] N is a reversible permutation matrix, which is defined by
- [ Pc ] 2 [ I ] 2
- ⁇ [ Pc ] N [ I N / 4 0 0 0 0 I N / 4 0 0 0 0 I N / 4 0 0 I N / 4 0 ] , N ⁇ 4. ( 5 )
- the matrix decomposition is the form of the matrix product of diagonal block-wise inverse Jacket sparse and Hadamard matrix
- [ C 2 0 0 B 2 ] is a diagonal block-wise inverse Jacket matrix, which has
- the permuted DCT-II matrix ⁇ tilde over (C) ⁇ N can be constructed recursively by using
- N 2 ⁇ N 2 DCT-II matrix, and [B] N/2 can be calculated by using
- [ B ] 2 [ C 8 1 C 8 3 C 8 3 - C 8 1 ] can be decomposed by using the 2-by-2 DCT-II matrix as
- ⁇ m is m-th angle (14)
- [ B ] N [ L ] N ⁇ [ C ] N ⁇ [ D ] N ⁇ ⁇
- ⁇ [ L ] N [ 2 0 0 ... - 2 2 0 ... 2 - 2 2 ... ⁇ ⁇ ⁇ ]
- [ B ] N [ C 4 ⁇ N ⁇ 0 C 4 ⁇ N ⁇ 1 C 4 ⁇ N ⁇ 2 ... C 4 ⁇ N ⁇ N - 1 C 4 ⁇ N ( 2 ⁇ k 0 + 1 ) ⁇ ⁇ 0 C 4 ⁇ N ( 2 ⁇ k 0 + 1 ) ⁇ ⁇ 1 C 4 ⁇ N ( 2 ⁇ k 0 + 1 ) ⁇ ⁇ 2 ... C 4 ⁇ N ( 2 ⁇ k 0 + 1 ) ⁇ ⁇ N - 1 C 4 ⁇ N ( 2 ⁇ k 1 + 1 ) ⁇ ⁇ 0 C 4 ⁇ N ( 2 ⁇ k 1 + 1 ) ⁇ ⁇ 1 C 4 ⁇ N ( 2 ⁇ k 1 + 1 ) ⁇ ⁇ 2 ... C 4 ⁇ N ( 2 ⁇ k 1 + 1 ) ⁇ ⁇ N - 1 ⁇ ⁇ ⁇ C 4 ⁇ N ( 2 ⁇ k N - 2 + 1 )
- FIG. 1A-1C is a butterfly data flow graph of Equation (27) according to an exemplary embodiment of the present invention. As shown in FIG. 1A-1C , the data flow of Equation (27) is represented.
- the DST-II matrix is decomposed based on the Jacket matrix by multiplying the same permutation matrices.
- the row permutation matrix is defined as
- Equation (30) we can obtain the 4 ⁇ 4 sparse matrix similar fashion the DCT-II of Equation (6) as follows:
- ⁇ [ A ] 2 [ S 8 1 S 8 1 S 8 3 - S 8 1 ] ( 31 )
- N ⁇ N DST-II matrix can be given by
- the N ⁇ N permuted DST-II matrix [S] N can be recursively formed by using
- a submatrix [A] N is given by
- Equation (36) The inverse of Equation (36) is simply calculated as below.
- submatrix [A] N can be represented by:
- Equation (38) is proved as follows.
- Equation (40) By taking Equation (40) and into the right hand side of Equation (38), the following equation is obtained.
- Equation (35) the left hand side of Equation (38) of [A] N from [S] 2N can be expressed by
- [ A ] N [ S 4 ⁇ N ( 2 ⁇ k 0 - 1 ) ⁇ ⁇ 0 S 4 ⁇ N ( 2 ⁇ k 0 - 1 ) ⁇ ⁇ 1 ... S 4 ⁇ N ( 2 ⁇ k 0 - 1 ) ⁇ ⁇ N - 1 S 4 ⁇ N ( 2 ⁇ k 1 - 1 ) ⁇ ⁇ 0 S 4 ⁇ N ( 2 ⁇ k 1 - 1 ) ⁇ ⁇ 1 ... S 4 ⁇ N ( 2 ⁇ k 1 - 1 ) ⁇ ⁇ N - 1 ⁇ ⁇ ⁇ ⁇ S 4 ⁇ N ( 2 ⁇ k N - 1 - 1 ) ⁇ ⁇ 0 S 4 ⁇ N ( 2 ⁇ k N - 1 - 1 ) ⁇ ⁇ 1 ... S 4 ⁇ N ( 2 ⁇ k N - 1 - 1 ) ⁇ ⁇ N - 1 ] ( 42 )
- Equation (41) and Equation (42) are the same and the expression of Equation (38) is correct.
- [ S ] N 2 N ⁇ [ Pr ] N ⁇ [ U N / 2 0 0 I N / 2 ] ⁇ [ I 2 ⁇ Pr N / 2 ] ⁇ [ I 2 ⁇ [ U N / 4 0 0 I N / 4 ] ] ⁇ [ I 4 ⁇ Pr N / 4 ] ⁇ [ I 4 ⁇ [ U N / 8 0 0 I N / 8 ] ] ⁇ ⁇ ... ⁇ [ I N / 4 ⁇ Pr 4 ] ⁇ [ I N / 4 ⁇ [ U 2 0 0 I 2 ] ] ⁇ [ I N / 2 ⁇ S 2 ] ⁇ [ I N / 4 ⁇ [ D 2 0 0 I 2 ] ] ⁇ [ I N / 4 ⁇ [ I 2 I 2 I 2 - I 2 ] ] ⁇ [ I N / 4 ⁇ Pc 4 ] ⁇ ⁇ ... ⁇ [ I 4 ⁇ [ D N / 8 0
- FIG. 2A-2C is a butterfly data flow graph of Equation (45) according to an exemplary embodiment of the present invention. As shown in FIG. 2A-2C , the data flow of Equation (45) is expressed.
- the DFT is a Fourier representation of a given sequence x(m),
- the DFT matrix [F] 4 can be represented as
- [ W ] N [ W 0 0 ... 0 0 W 1 0 ⁇ ⁇ ⁇ 0 ... 0 W N - 1 ]
- W is the diagonal complex unit for 2N point DFT matrix.
- FIG. 3A-3C is a butterfly data flow graph of Equation (56) according to an exemplary embodiment of the present invention.
- the data flow of Equation (56) is expressed as shown in FIG. 3A-3C .
- FIG. 4 is a butterfly data flow graph of Equation (64) according to an exemplary embodiment of the present invention.
- the data flow of Equation (64) is represented as shown in FIG. 4 .
- the DST-II computation can be from the computation of DCT-II by replacing the submatrix [L] N to [U] N .
- the DFT it computation can be from the computation of the DCT-II matrix by replacing the submatrix [D] N to [W] N , and the permutation matrix [L] N to [Pr] N as to the HWT, we not only need to replace submatrix [D] N by [Pb] N , and the permutation matrix [L] N by [Pa] N , but also multiply some special matrices at the output of HWT.
- FIG. 6A a simple generalized block diagram for the DCT-II/DST-II/DFT/HWT hybrid fast algorithms can be shown in FIG. 6A .
- FIGS. 6B and 6C shows variance comparison of the KLT/DCT/DST/DFT/Haar.
- the overall scheme switches between the directions as horizontal, vertical or bounding in intra mode depending on the prediction.
- the compare of computational complexity of conventional independent the DCT-II, DST-II, DFT, Haar transform and hybrid DCT-II/DST-II/DFT/HW are shown in Table I & II.
- the computational complexity of the proposed hybrid algorithms are slower than the conventional ones, due to the hybrid architecture algorithm.
- FIG. 5 is a block diagram of a signal transformation apparatus 500 adopting the hybrid architecture according to an exemplary embodiment of the present invention.
- FIGS. 6A and 6B are a detailed block diagram of the signal transformation apparatus 500 according to an exemplary embodiment of the present invention.
- the signal transformation apparatus 500 of FIG. 5 adopts the hybrid architecture for the DCT-II, the DST-II, the DFT, and the HWT.
- the signal transformation apparatus 500 includes a switching part 510 and a transformation part 520 .
- the switching part 510 selects one of the DCT-II, the DST-II, the DFT, and the HWT.
- the transformation part 520 transforms and outputs the input signal using the transform selected by the switching part 510 .
- the switching part 510 includes a submatrix selector 518 and a permutation matrix selector 519 .
- the transformation part 520 includes a first multiply 521 , a second multiply 522 , a third multiply 523 , a fourth multiply, and a fifth multiply 525 .
- the submatrix selector 518 selects the submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT.
- the permutation matrix selector 519 selects the permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT.
- the submatrix selector 518 selects [L] N 511 as the submatrix and the permutation matrix selector 519 selects [D] N 515 as the permutation matrix.
- the transformation part 520 computes Equation (18) using the submatrix [L] N and the permutation matrix [D] N and thus applies the DCT-II to the input signal. More specifically, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [L] N and the permutation matrix [D] N . Next, the second multiply 522 of the transformation part 520 multiplies the result by the matrix M.
- the transformation part 520 can transform the input signal using the DCT-II for the MPEG-4 HEVC.
- the submatrix selector 518 selects [U] N 512 as the submatrix and the permutation matrix selector 519 selects [D] N 515 as the permutation matrix.
- the transformation part 520 computes Equation (34) by using the submatrix [U] N and the permutation matrix [D] N and multiplying the matrix [M 1 ] [M 2 ] in each output, and thus applies the DST-II to the input signal. More specifically, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [U] N and the permutation matrix [D] N . Next, the second multiply 522 of the transformation part 520 multiplies the product by the matrix M. The third multiply 523 multiplies the matrix [M 1 ] [M 2 ] in each output of the previous product. Next, the transformation part 520 can generate the HEVC signal by applying the DST-II to the input signal.
- the submatrix selector 518 selects [Pr] N 513 as the submatrix and the permutation matrix selector 519 selects [W] n 516 as the permutation matrix.
- the transformation part 520 computes Equation (45) by using the submatrix [Pr] N and the permutation matrix [W] n , multiplying the input signal by the matrix [M 3 ] during the first h steps of the 2h steps, and multiplying the matrix [M 4 ] during the last h steps, and thus DFT-transforms the input signal.
- the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [Pr] N and the permutation matrix [W] n .
- the second multiply 522 of the transformation part 520 multiplies the product by the matrix M.
- the fourth multiply 524 multiplies the input signal by the matrix [M 3 ] during the first h steps of the 2h steps and multiplies the matrix [M 4 ] during the last h steps.
- the transformation part 520 can transform the input signal to the 3GPP, LTE, and DVB signal through the DFT.
- the submatrix selector 518 selects [Pa] N 514 as the submatrix and the permutation matrix selector 519 selects [Pb] N 517 as the permutation matrix.
- the transformation part 520 computes Equation (53) by using the submatrix [Pa] N and the permutation matrix [Pb] N and multiplying the input signal by the matrix [M 5 ] during the last h steps of the 2h steps, and thus HWT-transforms the input signal.
- the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [Pa] N and the permutation matrix [Pb] N .
- the second multiply 522 of the transformation part 520 multiplies the product by the matrix M.
- the fifth multiply 525 multiplies the input signal by the matrix [M 5 ] during the last h steps of the 2h steps.
- the transformation part 520 can transform the input signal to the JPEG-2000 signal through the HWT.
- the signal transformation apparatus 500 can selectively perform any one of the DCT-II, the DST-II, the DFT and the HWT by applying the hybrid architecture. Also, the signal transformation apparatus 500 can attain the hybrid architecture of low complexity by jointly applying the recursive structure of the DCT-II, the DST-II, the DFT, and the HWT.
- the present hybrid architecture exhibits the lower computational complexity than the direct scheme as shown in the following table.
- FIG. 7A is a diagram of a one-way video coding scenario according to an exemplary embodiment of the present invention.
- the video source encodes the video signal and broadcasts the signal to the TV or transmits the signal to the PC or the mobile display device via the server or the network.
- the video source may compress and provide the video as a DVD.
- the video source encodes the signal using one of the DCT-II, the DST-II, the DFT, and the HWT by use of the signal transformation apparatus 500 .
- the TV, the PC, and the mobile display device decode the signal using one of the DCT-II, the DST-II, the DFT, and the HWT with the signal transformation apparatus 500 .
- the signal transformation apparatus 500 can be applied to the one-way video coding scenario.
- FIG. 7B is a diagram of a two-way video coding scenario according to an exemplary embodiment of the present invention.
- the PC encodes the video signal captured by the camera and transmits the encoded video signal to the other PC.
- the PC receives and decodes the video signal of the other PC, and then displays the video on the display device. In so doing, the PC performs the encoding and the decoding using one of the DCT-II, the DST-II, the DFT, and the HWT with the signal transformation apparatus 500 .
- the signal transformation apparatus 500 can be applied to the two-way video coding scenario.
- the present invention can be applied to a signal transformation method of the signal transformation apparatus 500 .
- the present invention can be applied to a computer-readable recording medium containing a computer program for executing the functions of the signal transformation apparatus 500 .
- the present invention according to various embodiments can be embodied as a computer-readable code recorded to the computer-readable recording medium.
- the computer-readable recording medium can employ any data storage device for reading and storing data through the computer.
- the computer-readable recording medium can include ROM, RAM, CD-ROM, magnetic tape, floppy disc, optical disc, hard disc drive, and so on.
- a computer-readable code or program stored to the computer-readable recording medium may be transmitted over the network connected between computers.
- the signal transformation apparatus the signal transformation method, and the recording medium for select any one of the DCT-II, the DST-II, the DFT, and the HWT and transforming and outputting the input signal according to the selected transform
- the hybrid architecture of the low complexity can be attained.
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Abstract
Description
is its inverse, the matrix A corresponds to the Jacket matrix. The special sparse matrix belongs to the Jacket matrix.
can be seen as a special element-wise inverse matrix of
is permutation matrix. [Pr]N Permutation matrix is a special case, which has the form
and [Pc]N is a reversible permutation matrix, which is defined by
Where
the 2-by-2 DCT-II matrix and
Then we can write that it is clear that
is a diagonal block-wise inverse Jacket matrix, which has
where [C]N/2 denotes the
DCT-II matrix, and [B]N/2 can be calculated by using
can be decomposed by using the 2-by-2 DCT-II matrix as
is a lower triangular matrix,
is a diagonal matrix, and we use the cosine related function
cos(2k+1)φm=2 cos(2kφ m)cos φm−cos(2k−1)φm. where, φm is m-th angle (14)
and Φi=2i+1, ε{0,1,2, . . . ,N−1}. (16)
where ki=i+1, iε{0, 1, 2, . . . }.
C 4N (2k
where, m ε{0, 1, 2, . . . }.
−C 4N Φ
C 4N Φ
C 4N Φ
where Sl k=sin(kπ/l), ki=i+1, Φj=2j+1, i=0, 1, . . . , N−2, j=0, 1, . . . , N−1
and [U]N is an upper triangular matrix.
where ki=i+1, Φj=2j+1, i,j=0, 1, . . . , N−1.
[H] N =[H] 2 [H] N/2 (47)
for N=4, 8, 16, . . . and
For the remainder of this section, analysis will be concerned only with N=2k, k=1, 2, 3, . . . as the dimensionality of 2×2 Fourier matrices,
its inverse matrix is from element-wise inverse, such that
where [F]2=[{tilde over (F)}]2. The submatrix EN could be written by
[E] N =[Pr] N [{tilde over (F)}] N [W] N. (53)
and W is the diagonal complex unit for 2N point DFT matrix.
can be calculated by using
and the scaled coefficient: r=1√{square root over (2)}. Its inverse matrix is from element-wise inverse, such that
where is the Kronecker and ⊕ is the direct sum. For example, we calculate the dimension of Eq. (68)
(2h−1−1)×(N/2h +N/2h)+N/2h−1=(2h−1−1)×N/2h−1 +N/2h−1 =N−N/2h−1 +N/2h−1 =N.
Conventional methods | Proposed |
Addition | Multiplication | Addition | Multiplication | ||
DCT-II | Chen et al. | N (3log2 N + N − 1)/4 | (N2/32 + 2N)log2 N − N2/16 − N/2 |
3/2N (log2 N − 1) + 2 | N log2 N − 3/2N + 4 |
DST-II | Z. Wang | N (3log2 N + N − 1)/4 | (N2/32 + 2N)log2 N − N2/16 − N/2 |
3/2N (log2 N − 1) + 2 | N log2 N − 3/2N + 4 |
DFT | Cooley & Tukey | N log2 N | N (log2 N − 1/2) |
N log2 N | 1/2N (log2N) |
HWT | Andrews & Caspari | log2 N (2 + N/2) − N/2 | (3 + 3N/4)log2 N + 1 + N |
2(N − 1) | N | ||||
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KR20130042887A (en) | 2013-04-29 |
KR101362696B1 (en) | 2014-02-17 |
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